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Optimality principles have been used, in a holistic approach, to describe flow processes in several important geosystems. Optimality principles refer to the state of a physical system that is controlled by an optimal condition subject to physical and/or resource constraints.

While significant successes have been achieved in applying them, some principles appear to contradict each other.

For example, scientists have found that the formation of channel networks in a river basin follows the minimization of energy expenditure (MEE) rate, while the Earth-atmosphere system can be described by the maximum entropy production (MEP) principle.

Under isothermal conditions the energy expenditure rate is proportional to the entropy production rate; therefore, MEE and MEP do not appear to be consistent.

The physical origin of these optimality principles is an issue of active research. They cannot be directly deduced from existing thermodynamic laws that deal largely with processes within black-boxes (systems) and were not developed to describe flow structures for flow processes within these boxes.

The apparent inconsistency between different optimality principles calls for the development of a more precise understanding of fundamental physical laws within the context of thermodynamics.

In a recent article published in the Chinese Science Bulletin, Hui-Hai Liu, a scientist in the Earth Sciences Division at the Lawrence Berkeley National Laboratory of the University of California, proposed a new thermodynamic hypothesis.

In order to resolve the seemingly inconsistent optimality principles for flow processes in geosystems, this hypothesis states that a nonlinear natural system that is not isolated and involves positive feedback mechanisms tends to minimize its resistance to the flow process through it that are imposed by its environment.

The key discovery of this research is that a system does not tend to provide minimum resistance to all the involved flow processes, but only to the driving process imposed by its environment. The optimality principle corresponding to minimizing flow resistance applies solely to the driving process. This is a significant refinement of traditional optimality principles that do not single out the driving process.

This hypothesis resolves the seeming inconsistency between minimization of energy expenditure for a river basin and the maximum entropy production principle for the Earth-atmosphere system.

Water flow is the driving process in forming the channel network of a river basin; without water flow, there would not be a soil erosion process to generate river patterns.

On the other hand, the Earth receives radiation from the hot Sun and transfers this heat into space. The atmosphere and oceans act as a fluid system that transports heat from hot regions to cold ones with general circulation, and the convection process is more efficient in transferring heat than the conduction process. In this system, the driving flow process is the heat flow, which is also the initiator for other flow processes.

Under steady-state flow conditions, the average heat flow rate is closely related to entropy production in the Earth-atmosphere system, and the MEP corresponds to the maximum convective heat transport. In this case, maximum entropy production happens to be a byproduct of this heat-flow optimization process.

Observed and understood this way, the maximum entropy production principle in the Earth-atmosphere system and the minimization of energy expenditure in a river basin are consistent and can be unified in terms of minimizing resistance to the “flow process imposed by its environment”, or the driving process.

This research also outlines the conditions under which the corresponding optimality principle can apply, in a nonlinear system that is not isolated and involves positive feedback mechanisms.

Examples in subsurface liquid flow processes were used to demonstrate that the minimization of flow resistance does not hold when these conditions are not met.

This new hypothesis has important applications in practice.

Hui-Hai Liu posits that this new understanding can serve as the physical basis for successfully developing subsurface flow laws in hydrogeology, including the base-case theory for modeling unsaturated flow and transport in the well-known Yucca Mountain Project related to the US high-level nuclear waste repository site.

“I can see some direct applications of the theory in areas including fingering flow in the subsurface, hydraulic fracturing process, and rock damage mechanics,” said Hui-Hai Liu.